Donald Clayton Spencer (April 25, 1912 – December 23, 2001) was an Americanmathematician, known for work ondeformation theory of structures arising indifferential geometry, and onseveral complex variables from the point of view ofpartial differential equations. He was born inBoulder, Colorado, and educated at theUniversity of Colorado andMIT.

He wrote a Ph.D. indiophantine approximation underJ. E. Littlewood andG.H. Hardy at theUniversity of Cambridge, completed in 1939. He had positions at MIT andStanford before his appointment in 1950 atPrinceton University. There he was involved in a series of collaborative works withKunihiko Kodaira on thedeformation of complex structures, which had some influence on the theory ofcomplex manifolds andalgebraic geometry, and the conception ofmoduli spaces.

He also was led to formulate thed-bar Neumann problem, for the operator${\displaystyle \overline{\mathrm{\partial }}}$ (seecomplex differential form) in PDE theory, to extendHodge theory and then-dimensionalCauchy–Riemann equations to the non-compact case. This is used to show existence theorems forholomorphic functions.

He later worked onpseudogroups and their deformation theory, based on a fresh approach tooverdetermined systems of PDEs (bypassing the Cartan–Kähler ideas based ondifferential forms by making an intensive use ofjets). Formulated at the level of variouschain complexes, this gives rise to what is now calledSpencer cohomology, a subtle and difficult theory both of formal and of analytical structure. This is a kind ofKoszul complex theory, taken up by numerous mathematicians during the 1960s. In particular a theory forLie equations formulated byMalgrange emerged, giving a very broad formulation of the notion ofintegrability.

After his death, a mountain peak outside Silverton, Colorado was named in his honor.^[2]

• Kodaira–Spencer mapping
• Salem–Spencer set

• Schaeffer, A. C.; Spencer, D. C. (1950),Coefficient Regions for Schlicht Functions, American Mathematical Society Colloquium Publications, Vol. 35, Providence, R.I.:American Mathematical Society,ISBN 978-0-8218-1035-4,MR 0037908{{citation}}:ISBN / Date incompatibility (help)
• Schiffer, M. M.; Spencer, D. C. (1955),Functionals of Finite Riemann Surfaces,Princeton University Press^[3]
• Nickerson, H. K.; Spencer, D. C.; Steenrod, N. E. (1959),Advanced Calculus, Princeton, N.J.: Van Nostrand^[4]Nickerson, H. K.; Spencer, D. C.; Steenrod, Norman Earl (2011).Dover reprint.ISBN 978-0-4864-8090-9; pbk{{cite book}}: CS1 maint: postscript (link)
• Kumpera, A.; Spencer, D. C. (1972),Lie Equations: Volume I, General Theory, AM-73, Annals of Mathematical Studies, Princeton University Press,ISBN 978-0-6910-8111-3; pbk{{citation}}: CS1 maint: postscript (link)
• Kumpera, A.; Spencer, D. C. (1974),Systems of Linear Partial Differential Equations and Deformation of Pseudogroup Structures, Les Presses de l'Université de Montréal

• "Donald C. Spencer (1912–2001)"(PDF).Notices of the American Mathematical Society.51 (1). January 2004.
• Donald C. Spencer at theMathematics Genealogy Project
• O'Connor, John J.;Robertson, Edmund F.,"Donald Clayton Spencer",MacTutor History of Mathematics Archive,University of St Andrews
• "Donald C. Spencer, 89, Pioneering Mathematician, Dies".The New York Times. 1 January 2002.

• 1912 births
• 2001 deaths
• 20th-century American mathematicians
• Alumni of Trinity College, Cambridge
• Massachusetts Institute of Technology alumni
• National Medal of Science laureates
• People from Boulder, Colorado
• Princeton University faculty
• University of Colorado alumni

• Articles with short description
• Short description is different from Wikidata
• Articles with hCards
• CS1 errors: ISBN date
• CS1 maint: postscript